Equipartition of modal energy in an ideal string vibrating in the presence of a boundary obstacle

Abstract

We study the vibration characteristics of an ideal string in the presence of a curved boundary obstacle which is a common constructional feature of sitar, veena and tanpura. It has been well-established that these plucked string instruments have attractive tonal qualities like better harmonicity, presence of amplitude as well as frequency modulations, and appearance of a large number of overtones. The smooth wrapping and unwrapping of the string over the curved obstacle introduces quadratic nonlinearities whose effect on the vibration characteristics is elucidated through a perturbative analysis using the method of multiple scales. Modal interactions facilitated by the obstacle results in neutrally stable mode-locked periodic solutions in our model thereby accounting for the complex frequency and amplitude modulations. In particular, we find a mode-locked state which tends to an equipartition of energy with an increase in the number of modes. Our numerical simulations suggest that a plucked disturbance favors this state and hence, several overtones can be clearly identified in the sound of these plucked instruments. A good qualitative agreement between the results from our idealized model and the sound of these instruments highlights the role of the finite curved bridge in deciding the tonal quality of sitar, veena and tanpura.

Appendices

Appendix A: Discretized set of ODEs from Galerkin projections

For the discretization of our governing PDE (7), we assume the transverse displacement as

$$\begin{aligned} \displaystyle y(x,\tau )= & {} \alpha \gamma (b-\gamma )(1-x) \nonumber \\{} & {} + \sum ^{N}_{n=1}{\beta _{n}(\tau )\sin (n \pi x)}. \end{aligned}$$

(38)

Note that this assumed displacement profile trivially satisfies the boundary conditions in (8). Enforcement of the slope continuity boundary condition (9) gives us the wrapped length $\gamma$ in terms of the modal coordinates $\beta _n'$s as

$$\begin{aligned} \gamma = 1 - \sqrt{1-b+\frac{1}{\alpha }\sum ^{N}_{n=1}{n \pi \beta _n(\tau )}}. \end{aligned}$$

(39)

Substituting (38) and (39) in (7) and applying Galerkin projection, we get the discretized set of ordinary differential equations (ODEs) for the modal coordinates (see [3] for details) as

$$\begin{aligned}{} & {} \sum ^{N}_{n=1}\left[ K_{2nm}\left[ 1-b+\frac{1}{\alpha }\sum ^{N}_{p=1}{p\pi \beta _p}\right] \right. \nonumber \\{} & {} \quad \left. +\frac{n \pi }{2\alpha } \sum ^{N}_{p=1}{p\pi \beta _p \left( K_{4pm} - K_{3m}\right) }\right] \ddot{\beta }_n \nonumber \\{} & {} \quad - \frac{\left[ \sum \limits ^{N}_{p=1}{p\pi \dot{\beta }_p}\right] ^2}{4\alpha ^2 \left[ 1-b+\frac{1}{\alpha }\sum \limits ^{N}_{p=1}{p\pi \beta _p}\right] }\nonumber \\{} & {} \quad \left[ \sum ^{N}_{n=1}{n \pi \left[ 3 K_{4nm} + n \pi K_{1nm} - K_{3m}\right] \beta _n}\right. \nonumber \\{} & {} \quad +\left. 2 \alpha \left( 1-b\right) K_{3m}\right] \nonumber \\{} & {} \quad + \frac{1}{\alpha }\sum ^{N}_{p=1}{p \pi \dot{\beta }_p}\sum ^{N}_{n=1}{n \pi \dot{\beta }_n K_{4nm}} \nonumber \\{} & {} \quad + \sum ^{N}_{n=1}{n^2 \pi ^2 \beta _n K_{2nm}}=0, \end{aligned}$$

(40)

for $m=1,\, 2,...N$ and the $K'$s are defined as

$$\begin{aligned}{} & {} K_{1nm}=\int _0^1{(1-x)^2\sin (n\pi x)\sin (m\pi x) dx}\\{} & {} \quad = \left\{ \begin{array}{l l} \displaystyle \left( \frac{1}{6}-\frac{1}{4n^2\pi ^2}\right) \hspace{3mm}\text {if}\hspace{2mm} n=m,\\ \displaystyle \frac{4nm}{(n^2-m^2)^2\pi ^2} \hspace{5mm}\text {if} \hspace{2mm} n \ne m, \end{array}\right. \\ K_{2nm}= & {} \int _0^1{\sin (n\pi x)\sin (m\pi x) dx}= \frac{1}{2}\delta _{nm}, \\ K_{3m}= & {} \int _0^1{(1-x) \sin (m \pi x)dx}=\frac{1}{m \pi }, \\ K_{4nm}= & {} \int ^1_0{(1-x)\cos (n\pi x)\sin (m\pi x)dx}\\= & {} \left\{ \begin{array}{l l} \displaystyle \frac{1}{4n\pi } \hspace{12mm}\text {if}\hspace{2mm} n=m. \\ \displaystyle \frac{m }{(m^2-n^2)\pi } \hspace{5mm}\text {if}\hspace{2mm} n\ne m. \end{array}\right. \end{aligned}$$

These discretized ODEs are used to numerically study the various properties of the solution so as to verify the findings of the perturbative approach presented in this paper.

Table 7 Initial conditions for a ten mode approximation ($N=10$) corresponding to plucking at $x=x_2$

Appendix B: PDE at $\mathcal{O}(\epsilon ^2)$ resulting from (7)

$$\begin{aligned}{} & {} \frac{\partial ^2 y_2(x, T_0, T_1)}{\partial T_0^2}- \frac{1}{(1-\gamma _{st})^2}\frac{\partial ^2 y_2(x, T_0, T_1)}{\partial x^2} \nonumber \\{} & {} -\frac{2\gamma _1(T_0, T_1)}{(1-\gamma _{st})^3}\frac{\partial ^2 y_1(x, T_0, T_1)}{\partial x^2} \nonumber \\{} & {} \quad - \frac{(1-x)}{(1-\gamma _{st})}\frac{\partial ^2 \gamma _1(T_0, T_1)}{\partial T^2_0}\frac{\partial y_1(x, T_0, T_1)}{\partial x}\nonumber \\{} & {} \quad - \frac{2(1-x)}{(1-\gamma _{st})}\frac{\partial \gamma _1(T_0, T_1)}{\partial T_0}\frac{\partial ^2 y_1(x, T_0, T_1)}{\partial T_0\partial x} \nonumber \\{} & {} \quad + 2\frac{\partial ^2 y_1(x, T_0, T_1)}{\partial T_0\partial T_1}\nonumber \\{} & {} \quad - \frac{2(1-x)}{(1-\gamma _{st})^2}\frac{\partial y_{st}(x)}{\partial x}\left( \frac{\partial \gamma _1(T_0, T_1)}{\partial T_0}\right) ^2 \nonumber \\{} & {} \quad + \frac{(1-x)^2}{(1-\gamma _{st})^2}\frac{\partial ^2 y_{st}(x)}{\partial x^2}\left( \frac{\partial \gamma _1(T_0, T_1)}{\partial T_0}\right) ^2\nonumber \\{} & {} \quad -\frac{(1-x)\gamma _1(T_0, T_1)}{(1-\gamma _{st})^2}\frac{\partial y_{st}(x)}{\partial x}\frac{\partial ^2 \gamma _1(T_0, T_1)}{\partial T^2_0} \nonumber \\{} & {} \quad -\frac{2(1-x)}{(1-\gamma _{st})}\frac{\partial y_{st}(x)}{\partial x}\frac{\partial ^2 \gamma _1(T_0, T_1)}{\partial T_0 \partial T_1}\nonumber \\{} & {} \quad -\frac{(1-x)}{(1-\gamma _{st})}\frac{\partial y_{st}(x)}{\partial x}\frac{\partial ^2 \gamma _2(T_0, T_1)}{\partial T^2_0} \nonumber \\{} & {} \quad - \frac{\left( 3\gamma ^2_1(T_0,T_1) -2\gamma _{st}\gamma _2(T_0, T_1) +2\gamma _2(T_0, T_1)\right) }{(1-\gamma _{st})^4}\nonumber \\{} & {} \quad \times \frac{\partial ^2 y_{st}(x)}{\partial x^2}=0. \end{aligned}$$

(41)

Appendix C: Initial conditions corresponding to plucking at an arbitrary location along the string

An appropriate initial condition for the various modes of the string when plucked at a point corresponds to a triangular shape for the displaced string with the maximum displacement at the point of plucking. For the discretization of our system of equation in “Appendix A”, we have assumed the transverse displacement as given in (38). Multiplying (38) by $\sin (m \pi x)$ and integrating over the domain $x \in [0, 1]$ and using the orthogonality properties of the sine series, we get the modal coordinates in terms of the displaced profile $y(x,\tau )$ as

$$\begin{aligned}{} & {} \beta _m(\tau )=2\int _0^1{y(x,\tau )\sin (m\pi x) dx} \nonumber \\{} & {} \quad - \frac{2\alpha \gamma (b-\gamma )}{m \pi }. \end{aligned}$$

(42)

The initial shape for a string plucked at $x=x_2$ can be described through two line segments as

$$\begin{aligned}{} & {} y(x,0)=\alpha \gamma (b-\gamma ) - \frac{x}{x_2}[\alpha \gamma (b-\gamma ) - 1 + s], \nonumber \\{} & {} \quad \hspace{3mm} 0\le x \le x_2, \end{aligned}$$

(43)

$$\begin{aligned}{} & {} y(x,0) = 1-s +(1-s)\frac{x-x_2}{x_2-1}, \hspace{3mm} x_2\le x \le 1,\nonumber \\ \end{aligned}$$

(44)

where s is the downward displacement of the point $x=x_2$ measured from the horizontal line $y=1$ which is the nondimensional height of the bridge. Now the straight line (43) must have the same slope as the slope of the bridge at $x=0$. Substituting (43) into (9) and simplifying we get

$$\begin{aligned}{} & {} \alpha (2x_2-1)\gamma ^2 - \alpha (b x_2 + 2 x_2 -b) \gamma \nonumber \\{} & {} \quad + \alpha x_2 b -1 +s=0. \end{aligned}$$

(45)

Solving (45) we obtain the wrapped length of the string for the plucked profile as

$$\begin{aligned} \gamma \!=\!\frac{2 x_2 + b x_2 - b - \sqrt{(2 x_2 + b x_2 -b)^2 - \frac{4}{\alpha }(2 x_2 -1)(\alpha b x_2 - 1 +s)}}{2 (2 x_2 -1)}.\nonumber \\ \end{aligned}$$

(46)

Now substituting (43), (44) and (46) into (42) we can get the modal coordinates for any possible plucked (triangular) shape of the string. In Table 7 we have listed the initial modal coordinates (for $N=10$) for different plucking positions $x=x_2$ and $s=x_2+0.1$ which means the string is plucked approximately 0.1 nondimensional length units downward from the static configuration of the string. We note that the slope of the line corresponding to the static configuration of the string is $\alpha (b-2\gamma _{st}) (1-\gamma _{st})$ with $\gamma _{st}=1-\sqrt{1-b}$ which is approximately $-1$ but slightly lower than 1 in magnitude.

Appendix D: Slow flow equations for 3 modes in terms of amplitudes and relative phases

The slow flow equations obtained for the amplitudes and relative phases of a string with 3 modes ($N=3$) are

$$\begin{aligned}{} & {} \frac{\partial R_1}{\partial T_1} = \frac{\pi ^2}{2 \alpha (1-b)^{3/2}} \nonumber \\{} & {} \quad \left[ 3 R_2R_3\cos (\theta _2-\theta _1) + R_1 R_2 \cos \theta _1\right] , \end{aligned}$$

(47)

$$\begin{aligned}{} & {} \frac{\partial R_2}{\partial T_1} = \frac{\pi ^2}{8 \alpha (1-b)^{3/2}} \nonumber \\{} & {} \quad \left[ 6 R_1 R_3 \cos (\theta _2-\theta _1) - R^2_1\cos \theta _1\right] , \end{aligned}$$

(48)

$$\begin{aligned}{} & {} \frac{\partial R_3}{\partial T_1} = -\frac{\pi ^2}{2 \alpha (1-b)^{3/2}} \nonumber \\{} & {} \quad R_1 R_2\cos (\theta _2-\theta _1), \end{aligned}$$

(49)

$$\begin{aligned}{} & {} \frac{\partial \theta _1}{\partial T_1} = - \frac{\pi ^2}{ \alpha R_1 (1-b)^{3/2}}\nonumber \\{} & {} \quad \left[ 3 R_2 R_3 \sin (\theta _2-\theta _1) + R_1 R_2 \sin \theta _1\right] \nonumber \\{} & {} \quad + \frac{\pi ^2}{8 \alpha R_2 (1-b)^{3/2}}\nonumber \\{} & {} \quad \left[ 6 R_1 R_3 \sin (\theta _2-\theta _1) + R^2_1\sin \theta _1\right] , \end{aligned}$$

(50)

$$\begin{aligned}{} & {} \frac{\partial \theta _2}{\partial T_1}=- \frac{3 \pi ^2}{2 \alpha R_1 (1-b)^{3/2}}\nonumber \\{} & {} \left[ 3 R_2 R_3 \sin (\theta _2-\theta _1) + R_1 R_2 \sin \theta _1\right] \nonumber \\{} & {} +\frac{\pi ^2}{2 \alpha R_3 (1-b)^{3/2}}R_1 R_2 \sin (\theta _2-\theta _1). \end{aligned}$$

(51)